Finite Group Databasebeta

← #29665185

Subgroups — #29665185

Order 15616.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456741C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3#21455279C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445245C214yesmaximal, complemented
6#21445159C214yesmaximal, complemented
7#21444589C211yesmaximal, characteristic, complemented
8#21444503C211yesmaximal, characteristic, complemented
9C61×#1540C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C2×#14519777C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic
13#16840134C2212yescomplemented
14#16840114C2212yescomplemented
15#16840043—22no—
16#16840043C2214yes—
17#16839974—22no—
18#16839967—22no—
19#16839964C2212yes—
20#16839954C2212yes—
21#16839900—22no—
22C8×D244—22no—
23#16839827C2214yes—
24#16839815C2211yescharacteristic
25C2×#14519711—21no—
26#16839786C2211yescharacteristic
27#16839763—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839512C2214yescomplemented
31#16839509C2214yescomplemented
32C61×#532C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#332C2212yescomplemented
36C61×#331C2212yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40#14519802—44no—
41C2×#11758200—24no—
42C2×#11758201—22no—
43C2×#11758199—22no—
44C2×#11758194—21no—
45C2×C8×D61—24no—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777C2312yes—
50C4×D244—21no—
51C4×D244—21no—
52C4×D244D412yes—
53C4×D244C2312yes—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758198—21no—
58#14519711—24no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519700—24no—
64#14519699—24no—
65#14519698—22no—
66C22⋊#92418D412yes—
67#14519689—24no—
68C4×#92418C2312yes—
69C2×#41×C61—21no—
70C22×C8×C61—21no—
71C4⋊Q8×C61C2311yescharacteristic
72#127×C61C2311yescharacteristic
73#124×C61—21no—
74#121×C61—21no—
75#121×C61—21no—
76C4×D4×C61—21no—
77C4×D4×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61—21no—
83C4×C8×C61—21no—
84C23×D61—21no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—41no—
98C2×D244—21no—
99C2×D244—22no—
100C2×D244SD1614yescomplemented
101C2×D244C2×D412yes—
102C2×C4×D61—22no—
103C2×C4×D61—22no—
104C2×C4×D61—82no—
105C2×Dic122—22no—
106C2×Dic122—41no—
107C2×Dic122—82no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—42no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61C2411yescharacteristic
119#11758201—44no—
120#11758200—84no—
121#11758199—44no—
122#11758198—22no—
123#11758198—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129C4⋊Dic61C2×D412yes—
130#11758194—42no—
131C2×#92418—22no—
132C2×#92418C2×D414yes—
133C8×D61—84no—
134C23×C61—41no—
135C23×C61#12111yescharacteristic
136C22×D61—22no—
137C22×D61—24no—
138C22×D61—81no—
139C22×D61—81no—
140Q8×C61—41no—
141Q8×C61—41no—
142D4×C61—41no—
143D4×C61—41no—
144D4×C61—42no—
145D4×C61—81no—
146D4×C61—82no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—82no—
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61C2×SD1612yes—
155C2×C4×C61#13512yes—
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C2×#4111yescharacteristic
159C2×Dic61—22no—
160C2×Dic61—42no—
161C2×Dic61—81no—
162C2×Dic61—82no—
163D244—22no—
164D244—24no—
165D244C2×SD1614yes—
166C4×D61—44no—
167C4×D61—82no—
168Dic122—44no—
169Dic122—82no—
170C8×C61—42no—
171C8×C61—42no—
172#92418—24no—
173#92418—42no—
174#19385—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61#46512yes—
181C22×C61C2×#12111yescharacteristic
182C2×D61—24no—
183C2×D61—42no—
184C2×D61—42no—
185C2×D61—84no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—81no—
190C4×C61#46512yes—
191C4×C61C22×SD1611yescharacteristic
192C4×C61C2×#13511yescharacteristic
193C4×C61#57612yes—
194Dic61—42no—
195Dic61—82no—
196#3206—611no—
197C2×#474—611no—
198#3036—611no—
199#2972—611no—
200#1679—614no—
201#1665—614no—
202#1570—611no—
203#1549—611no—
204#1540D6111yescharacteristic, complemented
205C2×C61—22no—
206C2×C61—161no—
207C2×C61#304111yescharacteristic
208C2×C61C2×#46511yescharacteristic
209C2×C61#306911yescharacteristic
210D61—44no—
211C2×AGL(1,ℤ/8)—1221no—
212#532C2×D6111yescharacteristic
213C2×C4⋊Q8—611no—
214C2×C4×D4—611no—
215#492—612no—
216#487—612no—
217#474—614no—
218#474—1222no—
219#463—1222no—
220#461—612no—
221#461—1222no—
222#459—612no—
223#448—1222no—
224C8×D4—614no—
225C8×D4—1222no—
226#431—611no—
227#431C2×D6111yescharacteristic
228C2×#107—1221no—
229#423—611no—
230C2×C4⋊C8C2×D6111yescharacteristic
231#417—1221no—
232C2×#102—1221no—
233C2×C22⋊C8—1222no—
234#334—614no—
235#332—614no—
236#332C2×D6112yescomplemented
237#331C2×D6112yescomplemented
238C61#1938511yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×Q8—1221no—
241C22×D4—611no—
242C23×C4—1221no—
243AGL(1,ℤ/8)—2444no—
244C2×SD16—1224no—
245C2×SD16—1222no—
246C2×D8—1222no—
247C2×M16—1221no—
248C22×C8—21no—
249C22×C8—1224no—
250C4⋊Q8—612no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253C4⋊Q8—1221no—
254C4⋊Q8C22×D6111yescharacteristic
255#127C22×D6111yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—612no—
262C4×D4—612no—
263C4×D4—1221no—
264C4×D4—1221no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42C22×D6111yescharacteristic
270#107—1224no—
271C4⋊C8—21no—
272C4⋊C8—612no—
273C4⋊C8—612no—
274C4⋊C8—1221no—
275C4⋊C8C22×D6112yes—
276C4⋊C8C22×D6112yes—
277#103—1224no—
278#103—1222no—
279#102—1222no—
280#102—1224no—
281C22⋊C8—21no—
282C22⋊C8—612no—
283C22⋊C8—1224no—
284C4×C8—21no—
285C4×C8—612no—
286C24—1221no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—1222no—
291C2×Q8—4882no—
292C2×Q8—2441no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—612no—
298C2×D4—614no—
299C2×D4—1221no—
300C2×D4—1222no—
301C22×C4—41no—
302C22×C4—1221no—
303C22×C4—1222no—
304C22×C4—1222no—
305C22×C4—4882no—
306C22×C4—2441no—
307C22×C4#1175822211yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4D4×D6111yescharacteristic
310SD16—4884no—
311SD16—2444no—
312D8—2444no—
313M16—2442no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—42no—
317C2×C8—614no—
318C2×C8—1222no—
319C2×C8—4884no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—612no—
323C4⋊C4—1221no—
324C4⋊C4—1222no—
325C4⋊C4—1222no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—1222no—
331C22⋊C4—1222no—
332C42#1175822211yescharacteristic
333C42D4×D6111yescharacteristic
334C42D4×D6111yescharacteristic
335C42C23×D6111yescharacteristic
336C23—41no—
337C23—1224no—
338C23—1222no—
339C23—4881no—
340C23—4881no—
341C23#1451975011yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—4882no—
345Q8—2444no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—82no—
351D4—614no—
352D4—1224no—
353D4—1222no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—1222no—
359C2×C4—4882no—
360C2×C4—4882no—
361C2×C4—4881no—
362C2×C4—2442no—
363C2×C4—2444no—
364C2×C4#1451975011yescharacteristic
365C2×C4#1451975011yescharacteristic
366C2×C4#1451975011yescharacteristic
367C2×C4SD16×D6112yes—
368C2×C4#1451979612yes—
369C2×C4C2×D4×D6111yescharacteristic
370C2×C4C2×D4×D6111yescharacteristic
371C2×C4C2×#1175822211yescharacteristic
372C8—42no—
373C8—42no—
374C8—1224no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—1224no—
382C22—4884no—
383C22—2442no—
384C22—2442no—
385C22#1683999012yes—
386C22C2×#1451975011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—81no—
391C4—4882no—
392C4—2442no—
393C4#1683999012yes—
394C4C2×SD16×D6111yescharacteristic
395C4C2×#1451979611yescharacteristic
396C4#1684056412yes—
397C2—22no—
398C2—161no—
399C2—2444no—
400C2#2145532511yescharacteristic
401C2C2×#1683999011yescharacteristic
402C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#1540D61
hypocenterC61#19385
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21456741C2semidirect
2. C2×#16840043C2semidirect
4. #21454842C2semidirect
5. #21445245C2semidirect
6. #21445159C2semidirect
7. #21444589C2semidirect
8. #21444503C2semidirect
9. C61×#1540C2semidirect
13. #16840134C22semidirect
14. #16840114C22semidirect
30. #16839512C22semidirect
31. #16839509C22semidirect
35. C61×#332C22semidirect
36. C61×#331C22semidirect
100. C2×D244SD16semidirect
204. #1540D61semidirect
236. #332C2×D61semidirect
237. #331C2×D61semidirect
238. C61#19385semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214567412
C2×#168400432
#214552792
#214548422
#214452452
#214451592
#214445892
#214445032
C61×#15402
#1938561